Model with conditional interaction

In this example, we show regional effects of a model with conditional interactions using PDP, ALE, and RHALE. In particular, we:

1. show how to use effector to estimate the regional effects using PDP, ALE, and RHALE
2. provide the analytical formulas for the regional effects
3. test that (1) and (2) match

We will use the following model:

$$f(x_1,x_2,x_3)=-x_1^2{\mathbb{1}}_{x_2<0}+x_1^2{\mathbb{1}}_{x_2\ge 0}+e^{x_3}$$

where the features $x_1,x_2,x_3$ are independent and uniformly distributed in the interval $[-1,1]$.

The model has an interaction between $x_1$ and $x_2$ caused by the terms: $f_{1,2}(x_1,x_2)=-x_1^2{\mathbb{1}}_{x_2<0}+x_1^2{\mathbb{1}}_{x_2\ge 0}$. This means that the effect of $x_1$ on the output $y$ depends on the value of $x_2$ and vice versa. Therefore, there is no golden standard on how to split the effect of $f_{1,2}$ to two parts, one that corresponds to $x_1$ and one to $x_2$. Each global effect method has a different strategy to handle this issue. Below we will see how PDP, ALE, and RHALE handle this interaction.

In contrast, $x_3$ does not interact with any other feature, so its effect can be easily computed as $e^{x_3}$.